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Equilibrium and stability properties of a coupled two-component Bose–Einstein condensate

Chi-Yong Lin, E. J. V. de Passos, Mahir Saleh Hussein, Da-Shin Lee, Toledo Piza, A.F.R.de

Published Jan 1, 2003• 2 citations
OpenAlex

Abstract

The equilibrium and stability properties of a coupled two-component BEC is studied using a variational method and the one-dimensional model of Williams and collaborators. The variational parameters are the population fraction, translation and scaling transformation of the condensate densities, assumed to have a Gaussian shape. We study the equilibrium and stability properties as a function of the strength of the laser field and the traps displacement. We find many branches of equilibrium configurations, with a host of critical points. In all the cases, the signature of the onset of criticality is the collapse of a normal mode which is a linear combination of the out-of-phase translation and an in-phase breathing oscillation of the condensate densities. Our calculations also indicate that we have symmetry breaking effects when the traps are not displaced.

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The equilibrium and stability properties of a coupled two-component BEC is studied using a variational method and the one-dimensional model of Williams and collaborators. The variational parameters are the population fraction, translation and scaling transformation of the condensate densities, assumed to have a Gaussian shape. We study the equilibrium and stability properties as a function of the strength of the laser field and the traps displacement. We find many branches of equilibrium configurations, with a host of critical points. In all the cases, the signature of the onset of criticality is the collapse of a normal mode which is a linear combination of the out-of-phase translation and an in-phase breathing oscillation of the condensate densities. Our calculations also indicate that we have symmetry breaking effects when the traps are not displaced.
We study the interactions between two atomic species in a binary Bose-Einstein condensate to revisit the conditions for miscibility, oscillatory dynamics between the species, steady-state solutions, and their stability. By employing a variational approach for a quasi-one-dimensional, two-atomic species condensate, we obtain equations of motion for the parameters of each species: amplitude, width, position, and phase. A further simplification leads to a reduction of the dynamics into a simple classical Newtonian system where components oscillate in an effective potential with a frequency that depends on the harmonic trap strength and the interspecies coupling parameter. We develop explicit conditions for miscibility that can be interpreted as a phase diagram that depends on the harmonic trap's strength and the interspecies coupling parameter. We numerically illustrate the bifurcation scenario whereby nontopological, phase-separated states of increasing complexity emerge out of a symmetric state as the interspecies coupling is increased. The symmetry-breaking dynamical evolution of some of these states is numerically monitored and the associated asymmetric states are also explored.
The time evolution equations for average values of population and relative phase of a strongly coupled two-component Bose-Einstein condensate (BEC) are derived analytically. The two components are two hyperfine states, which are coupled by an external laser that drives fast Rabi oscillations between these states. Specifically, this derivation incorporates the two-mode model proposed in J. Williams et al., e-print cond-mat 9904399 for the strongly coupled hyperfine states $|1,\ensuremath{-}1〉$ and ∣2,1〉|2,1〉∣2,1〉 of 87Rb.{}^{87}\mathrm{Rb}.87Rb. The fast Rabi cycles are averaged out and the rate equations so derived represent the slow dynamics of the system. These include the collapse and revival of Rabi oscillations and their dependence on detuning and trap displacement as reported in experiments of J. Williams. A procedure for stabilizing vortices is also suggested.
We study the normal modes and the stability of two component condensates in a phase separated regime. In such a regime, the system can undergo a quantum phase-transition upon the change of interaction strength between bosons of the same species or the variation of the trap frequencies. In this transition, the distribution of the two components changes from a symmetric to an asymmetric shape. We discuss the nature of the phase transition, the role of the interface tension, and the phase diagram. The symmetric to asymmetric transition is the simplest quantum phase transition that one can imagine. We found new branches of normal modes which are specific for trapped multicomponent condensates and are analogous to the waves at the interface between two layers of immiscible fluids under gravity. At the point of the phase transition, the frequencies of those modes go to 0 and become imaginary which causes an instability. The interface tension shifts the normal-mode frequencies and changes the stability region of the system.
We study Bose-Einstein condensation (BEC) in three-dimensional two-component bosonic gases, characterizing the universal behaviors of the critical modes arising at the BEC transitions. For this purpose, we use field-theoretical (FT) renormalization-group (RG) methods and perform mean-field and numerical calculations. The FT RG analysis is based on the Landau-Ginzburg-Wilson ${\mathrm{\ensuremath{\Phi}}}^{4}$ theory with two complex scalar fields which has the same symmetry as the bosonic system. In particular, for identical bosons with exchange Z2{\mathbb{Z}}_{2}Z2​ symmetry, coupled by effective density-density interactions, the global symmetry is ${\mathbb{Z}}_{2,e}\ensuremath{\bigotimes}\mathrm{U}(1)\ensuremath{\bigotimes}\mathrm{U}(1)$. At the BEC transition, it may break into ${\mathbb{Z}}_{2,e}\ensuremath{\bigotimes}{\mathbb{Z}}_{2}\ensuremath{\bigotimes}{\mathbb{Z}}_{2}$ when both components condense simultaneously, or to $\mathrm{U}(1)\ensuremath{\bigotimes}{\mathbb{Z}}_{2}$ when only one component condenses. This implies different universality classes for the corresponding critical behaviors. Numerical simulations of the two-component Bose-Hubbard model in the hard-core limit support the RG prediction: when both components condense simultaneously, the critical behavior is controlled by a decoupled XY fixed point, with unusual slowly decaying scaling corrections arising from the onsite interspecies interaction.
In three-dimensional trapped Bose-Einstein condensate (BEC), described by the time-dependent Gross-Pitaevskii-Ginzburg equation, we study the effect of initial conditions on stability using a Gaussian variational approach and exact numerical simulations. We also discuss the validity of the criterion for stability suggested by Vakhitov and Kolokolov. The maximum initial chirp (initial focusing defocusing of cloud) that can lead a stable condensate to collapse even before the number of atoms reaches its critical limit is obtained for several specific cases. When we consider two- and three-body nonlinear terms, with negative cubic and positive quintic terms, we have the conditions for the existence of two phases in the condensate. In this case, the magnitude of the oscillations between the two phases are studied considering sufficient large initial chirps. The occurrence of collapse in a BEC with repulsive two-body interaction is also shown to be possible.
The dynamics of Josephson-like oscillations between two coupled Bose-Einstein condensates is studied using the time-dependent variational method. We suppose that the quantum state of the condensates is a Gaussian wave-packet which can translate and perform breathing shape oscillations. Under this hypothesis we study the influence of these degrees of freedom on the tunneling dynamics by comparing the full-model with one where these degrees of freedom are ``frozen'' at its equilibrium values. The result of our calculation shows that when the traps are not displaced the two models agree, whereas when they are, the models differ considerably, the former being now closer to its linear approximation.